Question

factor completely.

1. $6x^3 - 15x + 6$
2. $8x^3 + 4x^2 - 4x$
3. $12x^2 + 12x - 9$

Explanation:

Problem 1: Factor \(6x^2 - 15x + 6\) completely

Step 1: Factor out the GCF

First, find the greatest common factor (GCF) of \(6x^2\), \(-15x\), and \(6\). The GCF of \(6\), \(15\), and \(6\) is \(3\).

$$ 6x^2 - 15x + 6 = 3(2x^2 - 5x + 2) $$

Step 2: Factor the quadratic

Now, factor the quadratic \(2x^2 - 5x + 2\). We need two numbers that multiply to \(2 \times 2 = 4\) and add to \(-5\). The numbers are \(-1\) and \(-4\).
Rewrite the middle term:

$$ 2x^2 - 4x - x + 2 $$

Group the terms:

$$ (2x^2 - 4x) + (-x + 2) $$

Factor out the GCF from each group:

$$ 2x(x - 2) - 1(x - 2) $$

Factor out \((x - 2)\):

$$ (2x - 1)(x - 2) $$

Step 3: Combine the factors

Putting it all together, the factored form of \(6x^2 - 15x + 6\) is:

$$ 3(2x - 1)(x - 2) $$

Step 1: Factor out the GCF

Find the GCF of \(8x^3\), \(4x^2\), and \(-4x\). The GCF of \(8\), \(4\), and \(4\) is \(4\), and the GCF of \(x^3\), \(x^2\), and \(x\) is \(x\). So the GCF is \(4x\).

$$ 8x^3 + 4x^2 - 4x = 4x(2x^2 + x - 1) $$

Step 2: Factor the quadratic

Factor the quadratic \(2x^2 + x - 1\). We need two numbers that multiply to \(2 \times (-1) = -2\) and add to \(1\). The numbers are \(2\) and \(-1\).
Rewrite the middle term:

$$ 2x^2 + 2x - x - 1 $$

Group the terms:

$$ (2x^2 + 2x) + (-x - 1) $$

Factor out the GCF from each group:

$$ 2x(x + 1) - 1(x + 1) $$

Factor out \((x + 1)\):

$$ (2x - 1)(x + 1) $$

Step 3: Combine the factors

Putting it all together, the factored form of \(8x^3 + 4x^2 - 4x\) is:

$$ 4x(2x - 1)(x + 1) $$

Step 1: Factor out the GCF

Find the GCF of \(12x^2\), \(12x\), and \(-9\). The GCF of \(12\), \(12\), and \(9\) is \(3\).

$$ 12x^2 + 12x - 9 = 3(4x^2 + 4x - 3) $$

Step 2: Factor the quadratic

Factor the quadratic \(4x^2 + 4x - 3\). We need two numbers that multiply to \(4 \times (-3) = -12\) and add to \(4\). The numbers are \(6\) and \(-2\).
Rewrite the middle term:

$$ 4x^2 + 6x - 2x - 3 $$

Group the terms:

$$ (4x^2 + 6x) + (-2x - 3) $$

Factor out the GCF from each group:

$$ 2x(2x + 3) - 1(2x + 3) $$

Factor out \((2x + 3)\):

$$ (2x - 1)(2x + 3) $$

Step 3: Combine the factors

Putting it all together, the factored form of \(12x^2 + 12x - 9\) is:

$$ 3(2x - 1)(2x + 3) $$

Answer:

\(3(2x - 1)(x - 2)\)

Problem 2: Factor \(8x^3 + 4x^2 - 4x\) completely